In this paper, we study a class of constrained variational–hemivariational inequality problems with nonconvex sets which are star-shaped with respect to a certain ball in a reflexive Banach space. The inequality is a fully nonconvex counterpart of the variational–hemivariational inequality of elliptic type since it contains both, a convex potential and a locally Lipschitz one. Two new results on the existence of a solution are proved by a penalty method applied to a variational–hemivariational inequality penalized by the generalized directional derivative of the distance function of the constraint set. In the first existence theorem, the strong monotonicity of the governing operator and a relaxed monotonicity condition of the Clarke subgradient are assumed. In the second existence result, these two hypotheses are relaxed and a suitable hypothesis on the upper semicontinuity of the operator is adopted. In both results, the penalized problems are solved by using the Knaster, Kuratowski, and Mazurkiewicz (KKM) lemma. For a suffciently small penalty parameter, the solution to the penalized problem solves also the original one. Finally, we work out an example on the interior and boundary semipermeability problem that ilustrate the applicability of our results.

中文翻译：

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非凸星型集的约束变分-半变分不等式

在本文中，我们研究了一类带有非凸集的约束变分-半变分不等式问题，这些非凸集相对于自反Banach空间中的某个球是星形的。该不等式是椭圆型变分-半偏差不等式的完全不凸的对等物，因为它既包含凸电位又包含局部Lipschitz电位。通过将惩罚方法应用于由约束集距离函数的广义方向导数惩罚的变分-半变分不等式，证明了存在解的两个新结果。在第一个存在性定理中，假设了控制算子的强单调性和Clarke次梯度的松弛单调性条件。在第二个存在结果中，放宽这两个假设，并采用关于算子的上半连续性的适当假设。在这两个结果中，使用Knaster，Kuratowski和Mazurkiewicz（KKM）引理解决了惩罚性问题。对于足够小的惩罚参数，对受罚问题的解决方案也可以解决原始问题。最后，我们针对内部和边界半渗透性问题提出了一个例子，证明了我们的结果的适用性。